Kernkonzepte
The core message of this article is to investigate the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs. The authors establish several complexity results, including showing that recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, determining the precise complexity of recognizing 1-extendable (Es) graphs as Θp2-complete, and providing a linear-time algorithm to decide if a chordal graph is 1-extendable.
Zusammenfassung
The article focuses on the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs.
Key highlights:
Recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, resolving open problems.
Recognizing 1-extendable (Es) graphs is Θp2-complete, closing the complexity gap.
A linear-time algorithm is provided to decide if a chordal graph is 1-extendable, addressing an open question.
The complexity of recognizing well-covered triangle-free graphs and co-well-covered graphs remains open.
The authors first introduce the Wk hierarchy, where a graph G is Wk if for any k pairwise disjoint independent sets, there exist k pairwise disjoint maximum independent sets containing them. They show that recognizing Wk+1 graphs is coNP-complete, even when the input graph is Wk or Es.
Next, the authors investigate the complexity of recognizing Es graphs, where a graph is Es if every independent set of size at most s is contained in a maximum independent set. They prove that recognizing Es graphs is Θp2-complete, even when the input graph is Es-1.
For chordal graphs, the authors provide a linear-time algorithm to decide if a chordal graph is 1-extendable, by characterizing 1-extendable chordal graphs as those that can be partitioned into maximal cliques. They also show that recognizing Es chordal graphs is coW[2]-hard when parameterized by s.
The article concludes by highlighting two open problems: the complexity of recognizing well-covered triangle-free graphs and co-well-covered graphs.